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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2009

    saw activity at simple object and started a tiny section with examples.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2019

    Looking at this article: what is the need for having a zero object as opposed to just a terminal? In other words, would it be worse to say an object is simple if it has precisely two quotients, itself and 11?

    I’m particularly interested in the case of rings and commutative rings. One definition of simple ring is that it has precisely two two-sided ideals, and this condition is equivalent to the one I’m proposing. Similarly, a simple commutative ring under my proposal is just a field.

    (It may be that the concept becomes interesting only under additional assumptions, such as Barr-exactness or something in that vein. That’s why I phrased my question as I did: would my proposal be any worse than the one given?)

    • CommentRowNumber3.
    • CommentAuthorAli Caglayan
    • CommentTimeJul 31st 2019

    The definition of simple ring on the nlab is RR is simple as a bimodule. This seems to be equivalent to your definition. As far as I can tell your definition should subsume that definition of simple ring. I think the terminology becomes problematic for Lie algebras however.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 31st 2019

    Yeah, for Lie algebras there are practical reasons for excluding the 1-dimensional case.

    But I’m asking a more general theoretical question (insofar as there should be a general “theory” of simple objects in categories).

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeSep 7th 2023

    I beefed up Schur’s Lemma and proved it.

    diff, v20, current